Compensation gains are affected by PID in summing junctions so the variables must be independent.
Phase and frequency compensation may also need to include phase lead compensators with addition R in series with the integrator caps so improve stability at the closed loop unit gain margin or phase margin.
I made a simulator for this See comments.
These are not necessarily the best k factors for Kp,Ki,Kd.
One can plot / simulate a sig. gen. response of the PID filter.
[I did this]
For an intuitive time domain response consider this.
If you inject a slow triangle wave to all 3 Op Amps for gains \$k_p, k_i, k_d\$;
- the P amp just outputs a triangle
- the Derivative or D amp produces a square wave with Vpp/R=Ic=CdV/dt
- the Integral or I amp output almost a Sine wave but for DC is a steady ramp.
For a frequency response of a PID control consider this;
- The I response is an integrator with a -6dB/octave LPF slope like a Bass boost amp but integrates DC
- the D response has + 6dB/octave HPF slope like a treble boost amplifier
- the midband of the I and D filter results in a notch that shifts according to the I and D gains until you add the Proportional gain amp.
- the P Amp brings up the notch level of the midband and with sufficient gain flattens the midband entirely
- however in a closed loop system the PID is supposed to reduce the long term dc drift with the integrator, reduce HF noise with the gain of the D amplifier and reduce the midband error with high proportional gain.
- ultimately it depends on the inertia of the system, noise reduction, stability , step overshoot and slew rate desired for the plant or servo response desired and the power of the actuators, choice of feedback sensors and use of PID an other types of feedback that makes it possible to be stable.